Compactness and symmetric well-orders
Abstract
We introduce and investigate a topological form of Stäckel’s 1907 characterization of finite sets, with the goal of obtaining an interesting notion that characterizes usual compactness (or a close variant of it). Define a $T_2$ topological space $(X, \tau )$ to be Stäckel-compact if there is some linear ordering $\prec $ on $X$ such that every non-empty $\tau $-closed set contains a $\prec $-least and a $\prec $-greatest element. We find that compact spaces are Stäckel-compact but not conversely, and Stäckel-compact spaces are countably compact. The equivalence of Stäckel-compactness with countable compactness remains open, but our main result is that this equivalence holds in scattered spaces of Cantor–Bendixson rank $ \lt \omega _2$ under ZFC. Under $V=L$, the equivalence holds in all scattered spaces.
